when n is even, and
Q '11, ,, , if
-1
<~1> 2 (0402 (902
n~5 n~1
+ '<1>q>T+. . . +<-1>T£1>+q>"-1 when n is odd. If we put p=e's, g=e-'6, we obtain the formulae sin 710 =sin0 (2 cos0)"'1 - (n-2) (2 cos 0)"T“+ (2 cos0)"““ +(-1)' " '-I)( r;T' ' '( '2')<2 cos o>~~=~-1+ . (14) where n is any positive integer; I 2- z 2 c 2 2 2
(-1)Z 1sinn0=sin0 ncos0-n(n3,2>cos30+n(n 23101 4)cos5-. "
+(-If 1(2 cos 0)"" g (n even); (15) n-I
(-1) 2 sin n0=sin 6 1 -Q; cm20+£ cos“9- . . 1-1
+(-1)2 (2 cos 0)"" (rr odd). (16)
If we put in the same three formulae p=e'9, q= -e“'9, we obtain the series
n~2
(-1) 2 sinn0=cos6 sin"'10- (n-2)sin""'0-l- (n”3;$n)sin““50~... +(l)r S§ n»4~10+ (n even); (17)
n-l
(-1) 2 cos n0=the same series (rt odd); (18) 2 z
sin n0=cos 0 n sin 0- sin30+ s1n'6+ 2-1
+(-1)' (2 sin 0)"“1 (n even); (19) cos nl? =cos 0 I -£§ ¥sin'0+ sin'0-. . -l-(2 sin 9)"'1 (11 odd). (20)
We have thus obtained formulae for cos rc0 and sin 119 both in ascending and in descending powers of cos 0 and sin 0. Vieta obtained formulae for chords of multiple arcs in powers of chords of the simple or complementary arcs equivalent to the formulae (13) and (19) above. These are contained in his work T heoremala ad angular es section es. lacques Bernoulli found formulae equivalent to (12) and (13) (Mém. de l' Académie des Sciences, 1702), and transformed these series into a form equivalent to (10) and (11). lean Bernoulli published in the Acta erudilorum for 1701, among other formulae already found by Vieta, one equivalent to (17). These formulae have been extended to cases in which n is fractional, negative or irrational; see a paper by D. F. Gregory in Camb. Math. Joum. vol. iv., in which the series for cos rt0, sin 118 in ascending powers of cos 0 and sin 0 are extended to the case of a fractional value of n. These series have been considered b Euler in a memoir in the Nova acta, vol. ix., by Lagrange in his Calcul des functions (1806), and by Poinsot in Recherches sur Vanalyse des sections angulaires (1825).
24. The general definition of Napierian logarithms is that, if crfo'=a+¢b, then x+iy=log (a-l-tbl). Now we know that cx+'>'=exc0s y-l-tex sin y; ence ex cos y=a, ex sin y Zzeggfés =b, or e¢=(a2+b2)%, y=arc tan b/a=+=rn1r, where rn 3 'is an integer. If b=o, then rn must be even or odd according as a is positive or negative; hence log, (a-l-tb) =log, (a'+b2)%-l- L (arc tan b/aizrzvr) or log, (a-Hb) =log, (a'+b”)%-lf L (arc tan b/a=h2n+1r), according as a is positive or negative. Thus the logarithm of any complex or real quantity is a multiple-valued function, the difference between successive values being 21|'L; in particular, gzperbflk the most general form of the logarithm of a real positive me{°"° quantity is obtained by adding positive or negary five multiples of 2-m. to the arithmetical logarithm. On th1s subject, see De Morgan's Trigonometry artzl Double Algebra, ch. iv., and a paper by Professor Cayley in vol. 11. of Proc. London Malh. Soc.
25. We have from the definitions given in § 21, cos Ly-'= Hey-l-e-y) and sin ly=§ 1(ey-e-Lv). Theexpressions, § (ey-l-e-Lv), Hey-e-y) are said to define the hyperbolic cosine and sine of yand are written cosh y, sinh y; thus cosh y=cos ty, sinh y= -L sin ty. lhe functions cosh y, sinh y are connected with the rectangular hyperbola in a manner analogous to that in which the cosine and sine are connected with the circle. We may easily show from the definitions that
cos2(x-I-ty) -l-sin2(x-Hy) = 1,
cosh' y-sinh? y = I;
cos(x-|~iy) =cos x cosh y-i sin x sinh y, sin(x-l»-Ly) =sin x cosh y-l-1. cos x sinh y, cosh(a. -l-3) =C0sh a. cosh B-l-sinh a. sinh (3, sinh(a-l-3) =sinh a cosh, B-l-cosh a. sinh H. These formulae are the basis of a complete hyperbolic trigonometry. The connexion of these functions with the hyperbola was first pointed out by Lambert.
26. .If we equate the coefficients of rt on both sides of equation (13), this process requiring, however, a justification of its validity, we get I sinff 0 I 3 sins 0 I 3 S sin" 0 V Expansion = ' i. 4. . ofnnAngle
6 s1n0-l-2 3 -l-2 4 5 -I-2 4 6 7 -l-..., (2I)]npowers 0 must lie between the values is-ir. This equation Off” 5111* may also be written in the form
a s .1
arc sin x=x-P; 3%-l-Léi x
2 4 6
when x lies between =*= I.
By equating the coefficients of rt' on both sides of equation (12) we get
622. 2 § sm40 QSIDSB 2.4.6 s1nB0 U Sm H+; 2 +3-5 3 +5-5-7 4) 'l' " (22) which may also be written in the form A ~ 2:2 EE iff? 2-4-65
(arcsinx) x-l-3 2-l-3 5 3-l-3 5 7 4+... when x is between =*=1. Ditferentiating this equation with regard to x, we get
ar .'n x
if we put arc sin x=arc tan y, this equation becomes L Emi L4 2
arctany I, ' y, 1-l-3 I+y2+3 5 l | y, +... gfhis equation was given with two proofs by Euler in the Nova acta or 1793.
It can be shown that if mod x< I, then for any such real or complex value of x, a value of log, (1 +x) is given by the sum of the series xl-x2/2 -l-x3/3-.
We then have
~ (23)
1 1+x x3 x5 x7 g' »
5'°€r:¢-”+3+g+';+~—» sliillf”
put ty for x, the left side then becomes éllog (1-l-ty)-log (1 -f.y)] 01' I. 8.I'C tan y=|=Ln7|";
s s 1
hence arc tan yin-1r=y-;:;¢+%-3-§ +. .. The series is convergent if y lies between i 1; if we suppose arc tan y restricted to values between =b}1r, we have yfl 3,5
arc tan y=y-3—l-E-..., (24)
which is Gregory's series.
Various series derived from (24) have been employed to calculate the value of rr. At the end of the I7th century -fr was calculated to 72 places of decimals by Abraham Sharp, by means of the series obtained by putting arc tan y=1r/6, Seff" f°" y=I//3 in (24). The calculation is to be found in C"I"'""“°" Sherwin's Mathematical Tables (1742). About the same °f" time ]. Machin employed the series obtained from the equation 4 arc tan § -arc tan 2§ ;, =i1r to calculate 'll' to 100 decimal places. Long afterwards Euler employed the series obtained from § 1r=arc tan %-l- arc tan é, which, however, gives less rapidly converging series (Introd., Anal. infirz. vol. i.). T. F. de Lagny employed the formula arc tan 1//3=1r/6 to calculate rr to 127 places; the result was communicated to the Paris Academy in 1719. G. Vega calculated vr to 140 decimal places by means of the series obtained from the equation %r=5 arc tan 2.-I-2 arc tan 30. The formula § 1r=arc tan é-I-arc tan § -l-arc tan § was used 'by ]. M. Z. Dase to calculate vrto 200 decimal places. W. Rutherford used the equation -r=4 arc tan é - arc tan 710 -le arc tan 519. If in (23) we put y=% and %, we have 1r=8 arc tan %+4arc tan1}=2-4 I+;-~é~|-gil; 2 2 2 4 2 2
+56 1+ . + (wo) + §
3 100 3. 5 ' ' ' »
a rapidly convergent series for rr which was first given by Hutton in Plril. Trans. for 1776, and afterwards by Euler in Nova acta for 1793. Euler gives an, equation deduced in the same manner from the identity 1r=20 arc tan ~}-l-8 arc tan sag. The calculation of ir has been carried out to 707 places of decimals: see Proc. Roy. Soc. vols. xxi. and xxii.; also CIRCLE.
